Spherical geometry

Spherical geometry, and ball geometry or the geometry of the sphere engaged in points and point sets on the ball. Motivated originally selected by geometric considerations on the globe (see maps) and the celestial sphere (see astrometry ). Within the geometry, it is of particular interest, since they both represent a model of the elliptical geometry, with a suitable definition of the point on the sphere as also fulfills the axioms of projective geometry.

The spherical geometry strongly differs in some respects from the plane Euclidean geometry. It has no parallels, as two great circles, the analogue of the line on the ball, always intersect. Many of the well-known Euclidean geometry sets, such as the 180 ° angle sum in a triangle or the Pythagorean Theorem, are not valid on the sphere. There they are, however, in an adapted form.

• 2.1 lune
• 2.2 ball triangle

Basic concepts

The initial terms of planar geometries are the point and the line. On the ball these are defined as follows:

Straight

The role of the line comes to the great circles in the spherical geometry. Great circles are circles on the sphere whose ( Euclidean ) center is the center of the sphere. Examples of great circles on the globe are the equator and the meridians. A great circle obtained by the intersection of the sphere with a plane containing the ball center.

Point

Average of the ball with a Euclidean plane one obtains a circle. Is the distance of the center point of the ball to the plane intersecting the same as the radius of the sphere, the section of just a circle with a radius of 0, ie, a point on the sphere.

Geographical point

In the geographic view of a spherical geometry to define the point is taken from the Euclidean geometry, i.e., the amount of the spherical points is defined as the set of all points of the three-dimensional Euclidean space, which are located on the spherical surface.

Elliptical point

From the geometric point of view, the geographical definition of the point has a serious drawback. In geometric axiom systems is generally required that two points determine exactly one line. This is not the case in the above definition, if one considers for points on the sphere. Counter- points are points whose Euclidean connection line passes through the center of the sphere. ( So you relate to each other as north and south poles on the globe. ) By contrasting points run an infinite number of great circles (corresponding to the meridian on the globe ). Each great circle through a point also runs through his counter- point. It is therefore appropriate to combine pairs of counter points to a point.

Since the elliptical definition of the point identifies each point with its counter point, also each figure ( point set ) is identified on the ball with their counter-figure. (Eg particular, there is a triangle of two opposite triangles. )

Route

Distances are great circle arcs on the ball. The distance between two points A and B on the ball is the same as the length of the shortest major arc from A to B. On the unit sphere having the center M of the length of which is identical to the angle in radians. On a sphere with arbitrary radius R may be expressed as angular lengths. The actual spherical length d is then calculated from the angle in radians as.

In the definition of the elliptical spot, the smaller of the two angles between the connecting points of the counter corresponds to the spherical distance Euclidean line on the unit sphere. The distance is therefore never been greater.

Circle

Average of the ball with a Euclidean plane one obtains a circle. Of spherical geometry that is straight lines (cuts the ball with Euclidean planes containing the center of the sphere ) is nothing else than the other circuits. The cut circle of the sphere with a plane which does not contain the center of the sphere, small circle is called. (On the globe are, for example, with the exception of the equator, all latitudes small circles. )

Area calculation

Lune

Two great circles with the intersection points P and P ' divide the sphere into four spherical Two corner. A lune is limited by two P and P ' connecting arcs of great circles. The area of ​​a Kugelzweiecks is related to the total surface of the sphere as its opening angle to full angle:

In particular we on the unit sphere

Spherical triangle

The surface area of a spherical triangle with angles and is calculated from its angles:

Because the surface area is always greater than zero, the sum of the three interior angles of a spherical triangle must be greater than ( or 180 °):

The excess of the sum of the angles of the angles of a Euclidean triangle is called the spherical excess. The spherical excess of a triangle is proportional to its surface area ( and on the unit sphere with the proportionality factor 1 even the same).

The ball, as projective plane, duality and polarity

The spherical geometry is a projective plane with the elliptical definition of the points. In projective geometry can dualize all blocks, i.e., the concepts of point and line are swapped ( hence, lengths and angles as shown in the table above ). On the ball even leaves her every line a dual point A, and conversely any point A its dual line a clearly rejected. For a circle we obtain the dual pair of points as intersections of the sphere with the plane passing through the center of the sphere perpendicular to the plane of the circle (see figure).

In the dualization the incidence of points and lines is preserved. Thus: If a point A lies on a straight line b, the dual to him Just a passing through the straight line b for dual point B. However, not only the incidence is preserved, but also angles and lengths overlap. The dimension d of the angle between two straight lines A and B corresponds ( on the unit sphere ) the measure of the distance d between the dual to the line points A and B.

→ This duality is a special correlation and indeed an elliptic, projective polarity. This is explained in the article correlation ( projective geometry) in more detail.

Coordinates

To create a coordinate system, one takes a first arbitrary great circle as the equator. Then, it selects a meridian as zero meridian and defines a rotation. Now you can measure the angle from the equator and the prime meridian and therefore to define each position on the ball clear. Circles of latitude are parallel to the equator, while meridians pass through the two poles.

Limiting case of rule

In calculations on the spherical surface of the principle that all formulas containing the sphere's radius and therefore take into account the absolute size, need to go for the limiting case in valid formulas of plane geometry is considered.